Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Similar expressions are used for tunneling off the dot, G i (n), where n is thenumber of excess electrons on the QD. T i (E) is the tunneling matrix elementacross junction i. We do not explicitly calculate T i (E) but, rather, assign eachjunction with a phenomenological ‘‘tunneling resistance’’ parameter [7], R ifT i (E) 1 . We take into account the dependence of T i (E) on applied bias due tothe reduction of the average tunneling barrier height using the WKB approximation[73,74]. The tunneling resistances thus determine the averagetunneling rate between the two junctions, R 1 /R 2 = G 2 /G 1 . D i and D d are thedensity of states in the electrode and the dot, respectively, E i and E d are thecorresponding Fermi levels whose relative positions after tunneling, [E d (n F 1)E i (n)], depend on n, C 1 , C 2 , and the level spectrum, and f(E) is the Fermifunction. The condition for resonant tunneling is the lineup of the Fermilevel of the dot after tunneling (on or off the dot), with the Fermi level of theouter electrode before tunneling. D d is taken to be proportional to a set ofbroadened discrete levels corresponding to the positions of the discrete energylevels.First, one determines the probability distribution of n, P(n), from thecondition that at steady state, the net transition rate between two adjacent QDcharging states is zero [56]:PðnÞ½G þ i ðnÞ þ Gþ 2 ðnÞŠ ¼ Pðn þ 1Þ½G i ðn þ 1Þ þ G 2 ðn þ 1ÞŠ : ð3ÞThe tunneling current is then calculated self-consistently fromIðVÞ ¼ e X nPðnÞ½G þ 2 ðnÞG 2 ðnÞŠ ¼ e X nPðnÞ½G 1 ðnÞ G þ 1 ðnÞŠ ð4ÞAs a working example for the purpose of our explanation, we assumethat our QD has a bandgap E g =1 eV, with two discrete states in the VB andCB. The ground states are twofold spin degenerate, whereas the excited stateshave a fourfold degeneracy (including spin). The spacings between the states,D VB and D CB , are 0.3 and 0.4 eV for the VB and CB, respectively. In thepresent simulation, for simplicity, we did not allow for simultaneous tunnelingthrough the VB and CB states. The I–V curves are calculated as describedearlier, and the tunneling conductance curves (dI/dV versus V), are obtainedby differentiation.We present in Fig. 3 several limiting situations, where the relevantfactors to describe the junction asymmetry are the ratios C 1 /C 2 and R 1 /R 2 . Inall of the curves, the current in the bandgap region around zero bias is suppressed.However, considerable differences can be seen in the peak structureafter the onset of the tunneling current. In Figs. 3a and 3b, C 1 /C 2 = 0.1 and,therefore, most of the bias drops at junction 1. In Fig. 3a, R 1 /R 2 = 0.1, and thecalculated dI/dV curve exhibits resonant tunneling accompanied by single-Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.